What is the integral of p(p+3)^5 ?

The integral of p(p+3)^5 is (p^7)/7+(5p^6)/2+18p^5+(135p^4)/2+135p^3+(243p^2)/2+C

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integral of p(p+3)^5

\int p (p + 3)^{5} d p

\frac{p ^{7}}{7} + \frac{5 p ^{6}}{2} + 18 p^{5} + \frac{135 p ^{4}}{2} + 135 p^{3} + \frac{243 p ^{2}}{2} + C

Solution steps

\int p (p + 3)^{5} d p

Expand

p (p + 3)^{5} : p^{6} + 15 p^{5} + 90 p^{4} + 270 p^{3} + 405 p^{2} + 243 p

= \int p^{6} + 15 p^{5} + 90 p^{4} + 270 p^{3} + 405 p^{2} + 243 p d p

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int p^{6} d p + \int 15 p^{5} d p + \int 90 p^{4} d p + \int 270 p^{3} d p + \int 405 p^{2} d p + \int 243 p d p

\int p^{6} d p = \frac{p ^{7}}{7}

\int 15 p^{5} d p = \frac{5 p ^{6}}{2}

\int 90 p^{4} d p = 18 p^{5}

\int 270 p^{3} d p = \frac{135 p ^{4}}{2}

\int 405 p^{2} d p = 135 p^{3}

\int 243 p d p = \frac{243 p ^{2}}{2}

= \frac{p ^{7}}{7} + \frac{5 p ^{6}}{2} + 18 p^{5} + \frac{135 p ^{4}}{2} + 135 p^{3} + \frac{243 p ^{2}}{2}

Add a constant to the solution

= \frac{p ^{7}}{7} + \frac{5 p ^{6}}{2} + 18 p^{5} + \frac{135 p ^{4}}{2} + 135 p^{3} + \frac{243 p ^{2}}{2} + C

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What is the integral of p(p+3)^5 ?
The integral of p(p+3)^5 is (p^7)/7+(5p^6)/2+18p^5+(135p^4)/2+135p^3+(243p^2)/2+C